Finite Group Databasebeta

← #14532103

Subgroups — #14532103

Order 2176.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#63252C211yesmaximal, characteristic, complemented
2C2×#10587639C212yesmaximal, complemented
3#11762721C212yesmaximal, complemented
4#11762718C212yesmaximal, complemented
5#11762688C212yesmaximal, complemented
6#11762685C214yesmaximal, complemented
7#11762672C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C17C211yesmaximal, characteristic, complemented
9C25×C17C2211yescharacteristic
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C22×C22⋊D17C2214yes—
13C22×C22⋊D17C2212yes—
14C2×#63252—24no—
15C2×#63252—28no—
16C22×D68C2211yescharacteristic, complemented
17C22×C4×D17C2212yescomplemented
18C22×Dic34C2211yescharacteristic, complemented
19C2×C22⋊C4×C17—24no—
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊C4×C17C2212yes—
22#10587639—28no—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2214yes—
25C2×#63227C2214yescomplemented
26#10587502C2214yescomplemented
27#10587502C2212yescomplemented
28#10587502C2212yescomplemented
29C24×C17—22no—
30C24×C17—24no—
31C24×C17—24no—
32C24×C17—42no—
33C24×C17C2312yes—
34C24×C17C2311yescharacteristic
35C23×D17C2312yes—
36C22×C4×C17—24no—
37C22×C4×C17—22no—
38C22×C4×C17—42no—
39C22×C4×C17D414yescomplemented
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—28no—
46C2×C22⋊D17—28no—
47C2×C22⋊D17—44no—
48C2×C22⋊D17D418yescomplemented
49C22×Dic17C2314yes—
50C22×Dic17C2312yes—
51#63252—232no—
52C2×D68—24no—
53C2×D68—22no—
54C2×C4×D17—24no—
55C2×C4×D17—24no—
56C2×C4×D17—24no—
57C2×Dic34—22no—
58C2×Dic34—24no—
59C22⋊C4×C17—24no—
60C22⋊C4×C17—24no—
61C22⋊Dic17—24no—
62C22⋊Dic17—28no—
63#63227—44no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—28no—
67C23×C17—24no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—24no—
71C23×C17—416no—
72C23×C17—44no—
73C23×C17—44no—
74C23×C17C2×D414yes—
75C23×C17C2×D412yes—
76C23×C17#4114yescomplemented
77C23×C17C2411yescharacteristic
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—24no—
81C22×D17—24no—
82C22×D17C2×D414yes—
83C2×C4×C17—28no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—24no—
88C2×C4×C17—44no—
89C2×C4×C17C2×D412yes—
90C2×C4×C17C2×D412yes—
91C22⋊D17—216no—
92C22⋊D17—416no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—24no—
96C2×Dic17—24no—
97C2×Dic17—24no—
98C2×Dic17C2×D414yes—
99D68—28no—
100C4×D17—216no—
101Dic34—28no—
102#2404—171nomaximal, Sylow
103C22×C17—22no—
104C22×C17—22no—
105C22×C17—28no—
106C22×C17—24no—
107C22×C17—28no—
108C22×C17—416no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17—44no—
112C22×C17#12014yes—
113C22×C17C22×D412yes—
114C22×C17C22×D411yescharacteristic
115C22×C17C2×#4112yes—
116C22×C17H(5,𝔽2)12yes—
117C2×D17—24no—
118C2×D17—28no—
119C2×D17—42no—
120C2×D17—42no—
121C2×D17—44no—
122C4×C17—22no—
123C4×C17—22no—
124C4×C17—42no—
125Dic17—24no—
126Dic17—44no—
127C22×#41—171no—
128C2×#120—172no—
129C22×C22⋊C4D1711yescharacteristic, complemented
130#393—172no—
131#392—172no—
132#386—172no—
133#386—174no—
134#381—171no—
135C2×C17—24no—
136C2×C17—44no—
137C2×C17C2×#12011yescharacteristic
138C2×C17#52511yescharacteristic
139C2×C17#53511yescharacteristic
140C2×C17#54614yes—
141D17—44no—
142C25C2×D1711yescharacteristic
143C2×#41—348no—
144C2×#41—344no—
145C22×Q8—171no—
146C22×D4—172no—
147C22×D4—174no—
148C22×D4—171no—
149C23×C4—172no—
150C23×C4C2×D1711yescharacteristic
151C23×C4C2×D1711yescharacteristic
152#120—348no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—172no—
155C2×C22⋊C4—174no—
156C2×C22⋊C4—174no—
157C2×C22⋊C4C2×D1712yes—
158C2×C22⋊C4C2×D1712yes—
159#95—172no—
160#95—172no—
161#95—174no—
162C17#240411yescharacteristic, complemented, Sylow
163C24—24no—
164C24—22no—
165C24—24no—
166C24—42no—
167C24—172no—
168C24C22×D1712yes—
169C24C22×D1711yescharacteristic
170#41—3432no—
171C2×Q8—344no—
172C2×Q8—342no—
173C2×D4—178no—
174C2×D4—344no—
175C2×D4—344no—
176C2×D4—344no—
177C2×D4—348no—
178C2×D4—348no—
179C2×D4—342no—
180C2×D4—684no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—172no—
185C22×C4—174no—
186C22×C4—344no—
187C22×C4—344no—
188C22×C4—344no—
189C22×C4C22⋊D1714yescomplemented
190C22×C4C22×D1712yes—
191C22×C4C22×D1711yescharacteristic
192C22×C4C22×D1711yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—348no—
196C22⋊C4—344no—
197C22⋊C4—684no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—174no—
209C23—344no—
210C23—342no—
211C23—344no—
212C23—342no—
213C23#6325214yescomplemented
214C23D4×D1714yes—
215C23C2×C22⋊D1712yes—
216C23C23×D1711yescharacteristic
217Q8—348no—
218D4—348no—
219D4—3416no—
220D4—6816no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—28no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—174no—
228C2×C4—3416no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—344no—
232C2×C4—344no—
233C2×C4—344no—
234C2×C4C2×C22⋊D1712yes—
235C2×C4C2×C22⋊D1712yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—24no—
240C22—28no—
241C22—416no—
242C22—42no—
243C22—42no—
244C22—44no—
245C22—344no—
246C22—348no—
247C22—682no—
248C22—682no—
249C22—684no—
250C22#1058762314yes—
251C22C2×#6325212yes—
252C22C2×D4×D1712yes—
253C22#1058769112yes—
254C22C22×C22⋊D1711yescharacteristic
255C4—22no—
256C4—22no—
257C4—42no—
258C4—344no—
259C4—684no—
260C2—24no—
261C2—44no—
262C2—684no—
263C2C2×#1058762311yescharacteristic
264C2#1176332011yescharacteristic
265C2#1176337611yescharacteristic
266C2#1176344914yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C22⋊C4×C17C2
hypercenterC22×C22⋊C4D17
hypocenterC17#2404
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#63252C2semidirect
2. C2×#10587639C2semidirect
3. #11762721C2semidirect
4. #11762718C2semidirect
5. #11762688C2semidirect
6. #11762685C2semidirect
7. #11762672C2semidirect
8. C22×C22⋊C4×C17C2semidirect
16. C22×D68C22semidirect
17. C22×C4×D17C22semidirect
18. C22×Dic34C22semidirect
25. C2×#63227C22semidirect
26. #10587502C22semidirect
27. #10587502C22semidirect
28. #10587502C22semidirect
39. C22×C4×C17D4semidirect
48. C2×C22⋊D17D4semidirect
76. C23×C17#41semidirect
129. C22×C22⋊C4D17semidirect
162. C17#2404semidirect
189. C22×C4C22⋊D17semidirect
213. C23#63252semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#632522
C2×#105876392
#117627212
#117627182
#117626882
#117626852
#117626722
C22×C22⋊C4×C172
#240417